probabilty of mutually exclusive events and trouble with understanding notation

Now generalise to n mutually exclusive and exhaustive events ,... where

,

for ,..., and . What condition on must hold?

Note i(mod n) is the remainder when i is divided by n.

My problems:

- why the events are called not only 'mutually exclusive' but also 'exhaustive'? What does 'exhaustive' add to 'mutually exclusive'? Does it mean the three events 'take up' the whole sample space?

- what condition must hold on ? I know that , and the sum of includes at least this amount (1) and more (because and are included twice. So far I can only say that sum of is bigger than 1.

- I have trouble understanding the 'big cup' notation and how does it relate to the example with three As in the beginning of the question. If I cannot understand the notation how can I start thinking of answering the question... It looks like probability of a union of events , with i running from 0 to n-1 (i=r), but where does k come from and what is its role?